A227568 Largest k such that a partition of n into distinct parts with boundary size k exists.

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Author

Alois P. Heinz, Jul 16 2013

Keywords

• nonn

Comments

The boundary size is the number of parts having fewer than two neighbors.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Where records occur: A077043.

Cf. A227345, A227551.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, 1, 0),
      `if`(i<1, 0, max(`if`(t>1, 1, 0)+b(n, i-1, iquo(t, 2)),
      `if`(i>n, 0, `if`(t=2, 1, 0)+b(n-i, i-1, iquo(t, 2)+2)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t > 1, 1, 0], If[i < 1, 0, Max[If[t > 1, 1, 0] + b[n, i - 1, Quotient[t, 2]], If[i > n, 0, If[t == 2, 1, 0] + b[n - i, i - 1, Quotient[t, 2] + 2]]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

Formula

a(n) = max { k : A227345(n,k) > 0 } = max { k : A227551(n,k) > 0 }.

a(n) = floor(2*sqrt(n/3)).